Magnetism

Magnetism

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Magnetism is the phenomenon produced by moving charges (currents) and the intrinsic magnetic moments of electrons. The magnetic field B (unit: tesla) exerts a Lorentz force on charges and currents, and a torque on magnetic dipoles. Two foundational laws compute B from steady currents: the Biot–Savart law (dB = (μ₀/4π) (I dl × r̂)/r²) for arbitrary current elements, and Ampère’s circuital law (∮ B·dl = μ₀ I_enc) for highly symmetric geometries. Inside a long solenoid B = μ₀ n I, and inside a toroid B = μ₀ N I/(2πr). A charge q moving with velocity v in a uniform B executes circular motion of radius r = mv/(qB) and period T = 2πm/(qB), independent of speed — the basis of the cyclotron. For JEE Main, expect one or two problems from this cluster (≈ 3% weightage), usually combining Biot–Savart with Lorentz force or moving-coil galvanometer logic.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Sources and Description of B

A magnetic field is generated by (i) conduction currents in wires, (ii) moving charges, and (iii) the spin and orbital angular momenta of electrons in matter. Field lines are continuous closed loops — magnetic monopoles do not exist, so the net flux through any closed surface vanishes (Gauss’s law for magnetism).

Biot–Savart and Ampère’s Law

For a small current element I dl at position r from the field point:

dB = (μ₀/4π) (I dl × r̂)/r², with μ₀ = 4π × 10⁻⁷ T·m/A.

This yields the on-axis field of a circular loop of radius R at distance x from centre:

B = μ₀ I R² / [2 (R² + x²)^(3/2)]

For symmetric current distributions (infinite straight wire, solenoid, toroid), Ampère’s law is faster: ∮ B·dl = μ₀ I_enc.

Force, Torque and Flux

• Force on a current element: dF = I dl × B
• Force per unit length between two parallel wires carrying I₁, I₂ separated by d: F/L = μ₀ I₁ I₂ / (2π d) (attractive if currents are parallel)
• Lorentz force on charge: F = q(v × B + E)
• Torque on a magnetic dipole of moment m: τ = m × B; potential energy U = −m·B
• Flux: Φ = B·A; induced emf ε = −dΦ/dt (Faraday’s law)

Solenoid, Toroid and Galvanometer

• Inside an ideal solenoid: B = μ₀ n I, where n = turns per unit length
• Inside a toroid: B = μ₀ N I / (2πr)
• A moving-coil galvanometer uses the torque N I A B on a coil in a radial field; current sensitivity ∝ NBA/k

Exam Patterns

JEE Main typically tests: numerical evaluation of B on the axis of a loop, force per unit length between wires, radius/period of a charged particle, and identification of field direction using the right-hand rule. Conceptual questions on Earth’s magnetism (declination, dip, horizontal component B_H = B cos δ) also appear.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Helical Motion and the Cyclotron

When v has a component parallel to B, the perpendicular component produces circular motion while the parallel component is uniform, giving a helix of pitch v∥·T = 2π m v∥/(qB). The radius depends on speed but the period depends only on m/q and B, so all non-relativistic particles in uniform B complete a cycle in the same time regardless of energy — the key to the cyclotron, where an alternating electric field of fixed frequency f = qB/(2πm) accelerates the particle on each crossing. Above ~MeV energies the mass increase forces a switch to a synchrotron.

Edge Cases and Common Traps

• Right-hand rule direction: cross product I dl × r̂ is the most-skipped step; an inverted sign costs full marks.
• B due to a finite wire (not infinite) requires the full Biot–Savart integral: B = (μ₀ I)/(4π d) (sin θ₁ + sin θ₂).
• For a charge entering B at angle θ, radius r = m v sin θ/(qB) and pitch p = 2π m v cos θ/(qB) — many students forget the sin/cos split.
• Force between wires: attractive for like currents, repulsive for anti-parallel. Memorise F/L = μ₀ I₁ I₂/(2πd), not its sign convention.
• In a solenoid, B outside is ≈ 0 only for an infinitely long coil; finite solenoids have fringing fields.

Magnetic Materials

Diamagnets (χ < 0, weak, e.g. Cu, Bi) oppose the applied field. Paramagnets (χ > 0, small, e.g. Al) weakly align and follow Curie’s law χ ∝ 1/T. Ferromagnets (Fe, Ni, Co) exhibit hysteresis with a large, non-linear B–H loop characterised by retentivity and coercivity. The relative permeability μᵣ = 1 + χ distinguishes these classes numerically.

Worked Micro-Example

A proton (m = 1.67 × 10⁻²⁷ kg, q = 1.6 × 10⁻¹⁹ C) enters B = 0.5 T with v = 2 × 10⁶ m/s perpendicular to B. Radius r = mv/(qB) = (1.67 × 10⁻²⁷ × 2 × 10⁶)/(1.6 × 10⁻¹⁹ × 0.5) = 0.0418 m ≈ 4.2 cm. Period T = 2πm/(qB) = 1.31 × 10⁻⁷ s, independent of v.

Practice Prompts

1. A circular loop of radius 5 cm carries 4 A. Find B on the axis at 3 cm from the centre. (Answer uses B = μ₀ I R²/[2(R²+x²)^(3/2)] ≈ 1.6 × 10⁻⁴ T.)
2. Two long parallel wires 6 cm apart carry 3 A and 5 A in the same direction. Compute the force per unit length and state whether it is attractive or repulsive.

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